TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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A truncated Perron formula

Statement

If $F (s) = \sum a_{n} n^{- s}$ converges absolutely on $ℜ s = c > 0$ , then $∣ \frac{1}{2 π i} \int_{c - i T}^{c + i T} F (s) \frac{x^{s}}{s} d s - A^{*} (x) ∣ \leq \sum_{n \neq x} ∣ a_{n} ∣ (x / n)^{c} \min {1, (T ∣ \log (x / n) ∣)^{- 1}} + \frac{c ∣ a_{x} ∣}{T},$ where the final term is present only if $x \in Z_{> 0}$ .

Proof

Given: $c, T, x > 0$ and absolute convergence of the Dirichlet series at $c$ .

1.1

Absolute convergence permits termwise integration on the finite segment; subtracting the defining starred sum leaves $\sum_{n} a_{n} (I (x / n, T) - δ (x / n))$ .

givenalgebra

2.1

Apply the two branches of the truncated-kernel estimate termwise and the triangle inequality. The branch $n = x$ is precisely the displayed separate endpoint term.

step 1.1algebra∎

Depends on

Used by

The truncated von Mangoldt explicit formula Theorem

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Kiran S. Kedlaya, Analytic Number Theory, §10.2 (standard reference, not scraped)